1. Recovering metric and affine properties from images
Affine preserves:
Parallelism
Parallel length ratios
Similarity preserves:
Angles
Length ratios
Will show:
Projective distortion can be removed once image of line at
infinity is specified
Affine distortion removed once image of circular points is specified
Then the remaining distortion is only similarity

2. The line at infinity
For an affine transformation line at infinity maps onto line at infinity
The line at infinity l is a fixed line under a projective transformation H if and only if H is an affinity
A point on line at infinity is mapped to ANOTHER point on the line at infinity, not
necessarily the same point
Note: not fixed pointwise

3. Affine properties from images
projection
rectification
Euclidean plane
Two step process:
1.Find l the image of line
at infinity in plane 2
2. Transform l to its
canonical position
(0,0,1) T by plugging into
HPA and applying it to
the entire image to get a
“rectified” image
3. Make affine measurements
on the rectified image

4. Affine rectification v1 l∞ v2 l1 l3 l2 l4 c
a b

5. Distance ratios

6. The circular points
Two points on l_inf: Every circle intersects l_inf at circular points
“circular points” l∞
Circle:
Line at infinity

7. The circular points Canonical coordinates of circular points
Circular points are fixed under any similarity transformation
Canonical coordinates
of circular points
The circular points I, J are fixed points under the projective transformation H iff H is a similarity
Identifying circular points allows recovery of similarity properties i.e. angles
ratios of lengths

8. Conic dual to the circular points
The dual conic is fixed conic under the
projective transformation H iff H is a similarity
Note: has 4DOF (3x3 homogeneous; symmetric, determinant is zero)
l∞ is the nullvector

9. Angles Euclidean Geometry: not invariant under Projective:
projective transformation
Projective:
The above equation is invariant under projective transformation  can be applied
after projective transformation of the plane.
(orthogonal)
Once is identified in the projective plane, then Euclidean angles
may be measured by equation (1)

10. Length ratios

11. Metric properties from images
Upshot: projective (v) and affine (K) components directly determined from the image
of C*∞
Once C*∞ is identified on the projective plane then projective distortion may be
rectified up to a similarity
Can show that the rectifying transformation is obtained by applying SVD to
Apply U to the pixels in the projective plane to rectify the image up to a Similarity

12. SVD Decompose to get U Recovering up to a similarity from Projective
Perspective
transformation
Rectifying
Transformation: U
C*∞
Rectified image
Projectively
Distorted
Image
Euclidean
Similarity transformation

13. Recovering up to a similarity from Affine
transformation
Rectifying
Transformation: U
C*∞
Rectified image
Affinely
Distorted
Image
Euclidean
Similarity transformation

14. Metric from affine
Affine rectified

15. Metric from projective

16. Pole-polar relationship
The polar line l=Cx of the point x with respect to the conic C intersects the conic in two points. The two lines tangent to C at these points intersect at x

17. Correlations and conjugate points
A correlation is an invertible mapping from points of P2 to lines of P2. It is represented by a 3x3 non-singular matrix A as l=Ax
Conjugate points with respect to C
(on each others polar)
Conjugate lines
with respect to C*
(through each others pole)

18. Projective conic classification
Diagonal
Equation
Conic type
(1,1,1)
improper conic
(1,1,-1)
circle
(1,1,0)
single real point
(1,-1,0)
two lines
(1,0,0)
single line

19. Affine conic classification
ellipse
parabola
hyperbola

20. Chasles’ theorem A B C X D
Conic = locus of constant cross-ratio towards 4 ref. points A B C X D

21. Iso-disparity curves Xi Xj X1 X0 C1 C2 X∞

22. Fixed points and lines (eigenvectors H =fixed points)
(1=2  pointwise fixed line)
( eigenvectors H-T =fixed lines)

23. Projective 3D geometry

24. Singular Value Decomposition

25. Singular Value Decomposition
Homogeneous least-squares
Span and null-space
Closest rank r approximation
Pseudo inverse

26. Projective 3D Geometry Points, lines, planes and quadrics
Transformations
П∞, ω∞ and Ω ∞

27. 3D points 3D point in R3 in P3 projective transformation
(4x4-1=15 dof)

28. Planes 3D plane Transformation Euclidean representation
Dual: points ↔ planes, lines ↔ lines

29. Planes from points Or implicitly from coplanarity condition
(solve as right nullspace of )
Or implicitly from coplanarity condition

30. Points from planes (solve as right nullspace of )
Representing a plane by its span
M is 4x3 matrix. Columns of M are null space of
X is a point on plane
x ( a point on projective plane P2 ) parameterizes points on the plane π
M is not unique

31. Lines Line is either joint of two points or intersection of two planes
Representing a line by its span: two vectors A, B for two space points
(4dof)
Span of WT is the pencil
of points
on the line
Span of the 2D right
null space of W is the
pencil of the planes
with the line as axis
2x4
Span of W*T is the pencil of
Planes with the
line as axis
Dual representation: P and Q are planes; line is span of row space of W*
2x4
Example: X-axis:
join of (0,0,0) and (1,0,0) points Intersection of y=0 and z=0
planes

32. Points, lines and planes
Plane π defined by the join of the point X and line W is
obtained from the null space of M:
Point X defined by the intersection of line W with
plane is the null space of M

33. Plücker matrices L has rank 2
Plücker matrix (4x4 skew-symmetric homogeneous matrix)
L has rank 2
4dof
generalization of
L independent of choice A and B
Transformation
A,B on line AW*=0, BW*=0, …
4dof, skew symmetric 6dof, scale -1, rank2 -1
Prove independence of choice by filling in C=A+muB, AA-AA disappears, …
Prove transform based on A’=HA, B’=HB,…
Example: x-axis

34. Plücker matrices Dual Plücker matrix L* Correspondence
Join and incidence
(plane through point and line)
Indicate {1234} always present
Prove join (AB’+BA’)P=A (if A’P=0) which is general since L independent of A and B
Example intersection X-axis with X=1: X=counterdiag( )( )’=( )’
(point on line)
(intersection point of plane and line)
(line in plane)
(coplanar lines)

35. Plücker line coordinates
on Klein quadric
L consists of non-zero elements

36. Plücker line coordinates
(Plücker internal constraint)
(two lines intersect)
(two lines intersect)
(two lines intersect)

37. Quadrics and dual quadrics
Quadratic surface in P3 defined by:
(Q : 4x4 symmetric matrix)
9 d.o.f.
in general 9 points define quadric
det Q=0 ↔ degenerate quadric
(plane ∩ quadric)=conic
transformation
3. and thus defined by less points
4. On quadric=> tangent, outside quadric=> plane through tangent point
5. Derive X’QX=x’M’QMx=0
Dual Quadric: defines equation on planes: tangent planes π to
the point quadric Q satisfy:
relation to quadric (non-degenerate)
transformation

38. Quadric classification
Rank
Sign.
Diagonal
Equation
Realization 4
(1,1,1,1)
X2+ Y2+ Z2+1=0
No real points 2
(1,1,1,-1)
X2+ Y2+ Z2=1
Sphere
(1,1,-1,-1)
X2+ Y2= Z2+1
Hyperboloid (1S) 3
(1,1,1,0)
X2+ Y2+ Z2=0
Single point 1
(1,1,-1,0)
X2+ Y2= Z2
Cone
(1,1,0,0)
X2+ Y2= 0
Single line
(1,-1,0,0)
X2= Y2
Two planes
(1,0,0,0)
X2=0
Single plane
Signature sigma= sum of diagonal,e.g =2,always more + than -, so always positive…

39. Quadric classification
Projectively equivalent to sphere:
sphere
ellipsoid
hyperboloid of two sheets
paraboloid
Ruled quadrics:
hyperboloids of one sheet
Ruled quadric: two family of lines, called generators.
Hyperboloid of 1 sheet topologically equivalent to torus!
Degenerate ruled quadrics:
cone
two planes

40. Twisted cubic twisted cubic conic
3 intersection with plane (in general)
12 dof (15 for A – 3 for reparametrisation (1 θ θ2θ3)
2 constraints per point on cubic, defined by 6 points
projectively equivalent to (1 θ θ2θ3)
Horopter & degenerate case for reconstruction

41. Hierarchy of transformations
group transform distortion invariants properties
Projective
15dof
Intersection and tangency
Parallellism of planes,
Volume ratios, centroids,
The plane at infinity π∞
Affine
12dof
Similarity
7dof
The absolute conic Ω∞
Euclidean
6dof
Volume

42. Screw decomposition
Any particular translation and rotation is equivalent to a rotation about a screw axis and a translation along the screw axis.

43. 2D Euclidean Motion and the screw decomposition
screw axis // rotation axis

44. The plane at infinity
The plane at infinity π is a fixed plane under a projective transformation H iff H is an affinity
Represents 3DOF between projective and affine
canonical position
contains directions
two planes are parallel  line of intersection in π∞
line // line (or plane)  point of intersection in π∞

45. The absolute conic
The absolute conic Ω∞ is a (point) conic on π.
In a metric frame:
The absolute conic Ω∞ is a fixed conic under the projective transformation H iff H is a similarity
Represent 5 DOF between affine and similarity
Ω∞ is only fixed as a set, not pointwise
Circle intersect Ω∞ in two points
All Spheres intersect π∞ in Ω∞

46. The absolute conic Euclidean: Projective: (orthogonality=conjugacy)
Orthogonality is conjugacy with respect to Absolute Conic
d1 and d2 are lines

47. The absolute dual quadric
Absolute dual Quadric = All planes tangent to Ω∞
The absolute conic Q*∞ is a fixed conic under the projective transformation H iff H is a similarity
8 dof
plane at infinity π∞ is the nullvector of
Angles: